- ICES C:M. 1995
CM 1995/Q:10
FUNCTIONAL HETEROGENEITY:
USING THE INTERRUPTED POISSON PROCESS (IPP) MODEL UNIT
IN ADDRESSING HOW FOOD AGGREGATION MAY AFFECT FISH RATION
by
Jan E. Beyer
Danish Institute for Fisheries Research,
Charlottenlund Castle, DK-2920 Charlottenlund, Denmark
ABSTRACT
Addressing how aggregating and dispersing food organisms affect
fish ration, growth and survival is difficult without a
theoretical approach. The IPP building stone provides a
stochastic (individually-based), simple and well-defined
starting point.
Functional heterogeneity or how a predator perceives and
responds to patchy prey is quantified by considering the times
of prey-predator encounters as an interrupted Poisson process
(IPP) in the simplest case of one patch type and one prey type:
the predator locates food patches at random (1. parameter),
encounters prey in a Poisson process (2. parameter) when
foraging inside a patch, and leaves a patch at random (3.
parameter) .
The interencounter times become hyperexponentially distributed
(H 2 ) with parameters that are determined by the three
(biologically interpretable) parameters of the IPP. As an
example the model unit is applied to a cruising predator (e.g.
larval herring) with food patches (e.g. copepod nauplii) from
micro- to mesoscale and, maintaining a constant average prey
density (on a macroscale averaging over patch- and interpatch
volumes), the determination of the three IPP parameters is
discussed.
INTRODUCTION
The purpose of this note is to introduce the IPP and to
communicate the idea of using the IPP as a first step towards
developing a new theory for predator-prey encounters in case of
patchy prey (see OH1). This is done by using the overheads
(referred to as OH1 to OH6) as the core of this note as weIl.
Emphasis is on simplicity. However, even simple stochastic
processes have a tendency to reveal complex behaviour and utmost
care is needed to avoid misinterpretations and other pitfalls.
As such the present note only serves as an appetizer and the
interested reader is referred to Beyer and Nielsen (in press)
for a more thorough examination of the IPp l ).
The background for OH2 is the process of encountering prey by an
individual predator. The outcome is a signal comprising the
individual encounters, which may be specified by the interencounter times (T) or by the number of encounters (~=counts)
during a specific period of time (t); see Fig 1.
t
Or
1•
•2 •• • • •
••••
j
•• Nt
•••••
••
a
•~T-i••
• •
~
time
Fig 1.
Random encounters means that a predator encounters prey
particles in a Poisson process (PP) at some rate AoO, i.e. the
interencounter times are exponentially (EXP) distributed with
mean = standard deviation = l/A o and the number of counts is
Poisson distributed with mean = variance = Aot. Thus the
coefficient of variation (= standard deviation/mean) of the
interencounter times is one and so is IDC, the index of
dispersion for counts (= variance to mean ratio for ~), i.e.
c.v. (T) = IDC(t) = 1 for all t. It is suggested to use IDC as a
measure of the functional heterogeneity in encounters. We expect
IDC > 1 in case of patchy prey.
Rothschild 2l (1991) considers a simple patchy extension of Poisson
encounters by assuming the interencounter times to be described
by a mixed exponential distribution (= hyper-exponential (H 2 ) =
COMPOUND EXP in OH2). It is of note that the equivalence between
the exponential and the Poisson distribution does not extend to
the compound (or mixed) cases.
The three parameters of a H2 -distribution (which will be
introduced later) cannot be given a straightforward biological
1)
Beyer & Nielsen, 1995: Predator foraging in patchy
environments: the interrupted Poisson process (IPP)
model unit. Dana. In Press.
2)
Rothschild, B.J., 1991. Food-signal theory: population
regulation and the functional response. J. Plank. Res. Vol.
13 no. 5: 1123-1135.
2
•
, .. : "'!
I' ,
~'"
"
•.
. 'r,
-interpretation in .terms of patchiness._A basic point of this
"study iso there.fore that. the-H2-distribution. of int"erencounter
times does not (biologically) constitutes the starting point but
instead is derived as a consequence of the IPP-model, which is
specified by three biological parameters associated with the
foraging cycle, i.e. the rates at which the predator encounters
patches (W2)' leaves patches (Wl) and encounters prey particles
(A) while foraging inside patches. These three rates or
parameters of the IPP are considered constant. The only real
change," moving from Poisson to IPP encounters, is that the
process of random encounters is interrupted by exponentially
distributed periods of interpatch travel (during which the
predator is assumed not to encounter prey particles) .
EXAMPLE
•
A conceptual example is used to illustrate how the IPPparameters can be obtained from simple encounter theory and the
characteristics of patches. The simple scenario is shown in OH3
and comprises a cruising predator and a population of immobile
prey particles (e.g. a fish larva and copepod nauplii, which are
shown as dots).
Starting with randomly distributed prey in the water volume
unit, the predator encounters these particles at a rate Ao, which
equals the encounter rate kernel (= search volume rate) times
the average prey density (Po), i.e.
POISSON: prey encounter rate
assuming constant swimming speed (v) and that the perceptive
distance (d) is large compared to the radius of a prey particle
(and that possible effects of crossing pathways can be
neglected) .
•
In creating patchiness with the same number of prey particles
(mass balance), the patch volume fraction (~) determines the
prey density inside patches (p=Po/~) and, assuming unchanged
kerneI, the prey parameter of the IPP is obtained as
IPP: prey encounter rate
The concentration of patches (C) and the patch volume (V) are
inversely related:
=
C'V
Consider
assume a
actually
of patch
W2
~
= Pol P
a patch as a spherical particle with radius Rand
patch encounter to take place when the predator
encounters the patch surface then (assuming the chance
overlap is small)
=
3/4 v
~
IPP: patch encounter rate
IR;
A patch encounter represents the only entrance to the food
supply but (within the IPP concept)a predator may fail to
3
encounter"prey particles during· a (short) patch .residence. Even..
if the·predatorwas·able to detect a patch surface at distance d
the patch encounter rate above is not likely to change (because
d is considered considerably smaller than R). The implication of
W2 being inversely proportional to Rappears to be that a prey
population can reduce its chance of being predated by forming
bigger but fewer patches. However; it also depends on the
behaviour of the predator once a patch encounter has occurred.
Assuming that the patch volume fraction equals the fraction of
time a predator spends in patches, W2/ (W\+W2) (, see next
section), the mean patch residence time, l/w\, becomes equal to
the time required for swimming the distance 4/3R:
w\
=
0
W2' (1] t-
1 ) ... 1/4/3R/V ;
IPP: patch leaving rate
where the approximation is valid for small 1]. Thus the average
patch residence time will increase (as 1/w2' the mean interpatch
travel time) in direct proportion to an increasing patch
dimension, which will insure that the predator on average
encounters prey particles at exactly the same rate independently
of the degree patchiness (i.e. of 1] and R). This, of course, is
convenient for comparisons of the variability in the IPPcounting process with that of random PP-encounters. But in
reality it means that predator behaviour is neglected; a point
I'll return to in the conclusion.
•
MODEL
In an environment containing food patches a predator is
considered to alternate between two environmental states:
state 0: non-patch
state 1: patch
(i.e. interpatch travelling)
(i.e. foraging inside a patch)
The predator encounters patches at random (in a Poisson process)
at rate W2 and once an encounter has taken place the predator
also leaves the patch at random at rate wt ' i.e.the duration of
patch residences is exponentially distributed with mean l/wt. The
equilibrium probability distribution is given by
where n t may be interpreted as the proportion of time spent in
food patches when the foraging behaviour of an individual
predator is studied over a long period of time. An alternative
empirical interpretation of the equilibrium distribution is that
nt denotes the fraction of the total number of predators, which,
at some fixed (but arbitrary) point in time, is foraging inside
patches (assuming negligible effect of predatory interactions
during the period of timee considered).
The transition diagram of the complete IPP is shown on OH4. When
state 1 (patch) is occupied, the forager encounters prey items
in a Poisson process at a constant rate, A. This completes the
simple patch and prey IPP-model unit.
4
•
The IPP(A,WltWz) process is stochastically equivalent to a
Hz (p,)'lt)'z) renewal process )i.E:. with interencounter times being
Hz distributed - see Fig 2) and the (non-trivial) relationship
between the parameters is given by
A
P')'I
+
(l-p) "'12
)'1
•
+
)'2
The H2 -renewal process may be interpreted in the following way
assuming e.g. )'1>)'2' On each prey encounter, the forager decides
whether to stay in the patch or to leave: with probability p it
stays and the probability of encountering the next prey item in
that patch is then governed by the first exponential phase, i.e.
the interencounter time becomes exponentially distributed with
mean 1/)'1' With probability 1-p it leaves the patch and the time
required to encounter the next prey item (and hence a patch)
will be governed by the second exponential phase, i.e. the
interencounter time becomes exponentially distributed with mean
1/1'2 > 1/1'1' Note that the patch concept in this Hz-interpretation
is quite different from the more straightforward IPP-patch
concept.
EXP()'1J
time
mean
mean
•
-I-
L--
L-l-----=:::::=~time
weighting factor
weighting factor
-+-
.L..-_ _
---
- - - - ._-_-_._-_._--=-_-__--'-'-=-=-=~ time
.
Fig 2. A graphical (pdf) representation of the phase diagram
(see OH4) for a hyperexponential distribution with two phases
(Hz) also known as a mixed exponential distribution or a compound
exponential. Its pdf is given by the weighted sum of the two
exponential pdf' s, i. e. P"YlexP (-"Ylt) + (l-p) "Y2exP (-"Y2t) .
5
·RESULTS
Once the IPP is specified by the parameters (A,W\,W2) or by its
H2-parameters (P'~\'~2) properties of the counting process (and
thus the probability distributions governing the potential
ration) can be studied. The results will differ for the timestationary process and the event-stationary processes with
respect to transient behaviour. The process is in equilbrium or
time-stationary if it has been running a long time before
observations starts (t=O) or simply, if is being observed from
(t=O) a random point in time. The event stationary case implies
that an encounter has just taken place at t=O. In this note,
which serves as an IPP-introduction, focus is just on the mean
counts and on the index of dispersion for counts (IDC) defined
as the variance to mean ratio for counts. Only the asymptotic
IDC is given below, i.e. the limit IDC, which is valid for
counts in both the time- and event-stationary processes that
pertain to periods of time large compared to 1/(w\+w2).
The mean number of encounters in the time-stationary process is
~
mean count
This result was expected because At is the mean count in a
Poisson process but the predator is only occupying state 1
(patch) in the fraction n\ of the time t. The result is exact for
all t in the time-stationary case but only asymptotically
correct in the event-stationary case. The asymptotic IDC becomes
constant:
Consider this IDC (, which also equals the c.v. squared of the
interencounter time,) as a function of W\' the rate of leaving
patches for fixed rates of encountering patches (W2) and prey (A)
inside patchesi see OHS. If w\=O the predatory will stay in the
patch and thus continues to encounter prey particles in a
Poisson process so IDC takes the minimum value of one. At the
other extreme, considering very large w\' the predatory will
still encounter patches at a constant rate, W2' but it will leave
the patches so fast that the chance of encountering a prey is
negligible, which represents a special (and unrealistic) case of
Poisson so IDC=l for this extreme as weIl. The maximum (in
between these extremes) occurs whenthe predator leaves the
patches at the same rate it is encountering patches, i.e. for
w\ =W2 when n o=~,
IDC mu = 1 +
~A/W
That is the maximum IDC for the IPP exceeds the variance to mean
6
•
ratio. for the Poisson' process··.with Jh'A/w, ,half· the rate:ratio of
encountering prey in patches to encountering or leaving patches.
As an example consider a case,
'A
p = 8/11
IPP:
1'1= 4
1'2= 1/3
•
Wl=
W2=
3
8/9
4/9
=
in which the mean counts is one per unit of time and IDC=4. The
values of the IPP-parameters mayaIso be obtained using 'A o=l as
the starting point (see OH3) and with a patch volume factor of
~=3. The rate of leaving patches is double the rate of
encountering patches, W 1 = 2W2 (because ~·'-1=2), which coinsides
with the inflexion point on the IDC curve (OHS). The curve is
rather flat (i.e. with a very long right tail) giving a maximum
IDC of 4+3/8 when w, is reduced to 4/9 .
Based on this example OH6 shows a realization of the patchy IPP
(right), which can be compared with a realization of the nonpatchy Poisson process (left) specified by 'Ao=l, i.e. the mean
number of counts is one per unit of time in both cases. The top
graphs show number of counts per 100 time units. The average
count is thus 100 but the variability is much higher in the IPP
case (IDC=4i standard deviation = 20) than in the Poisson case
(IDC=li standard deviation = 10). The graphs below are zooming
in, magnifying 10% segments of the graphs above. IDC is still
ca. 4 for the mid-cases because t=10 is considerably bigger than
1/(w,+w2) = 3/4. OH6 (right) is based on a simulation of the H2
process.
CONCLUSION
•
The three parameters of the IPP model unit are related directly
to patch characteristics and foraging behaviour as illustrated
by the basic example in OH3. The patch leaving rate was
determined as W,=W2/~ by assuming the patch volume fraction to
equal the fraction of time spend by the predator in patches,
i.e. ~=n,. This implies that the predator on average encounters
prey particles at the same rate in different situations of
patchiness (specified by ~, the patch volume fraction and by R,
the patch radius) including the random or non-patch (Poisson)
situation for ~=1. However, in reality ~ is usually small, say,
in the order of 1/100 for larval fish predators, which means
first that the prey encounter rate may become very high and
secondly that the patch leaving rate also becomes high (to
ensure a constant mean encounter rate). In this case of a small
~ the mean count and the asymptotic index of dispersion for
counts become
mean counts per unit time
IDC'"
1 + 2· 'A/w,
= W2·'A/W,
i
7
The important· point is that the .mean' 'count 'and the TDC are
approximately proportional to A/Wl' the rate of encountering prey
particles divided by the rate of leaving patches both of which
are proportional to the average swimming speed of the predator
when it is foraging inside a patch (denoted by, say, VIpred where
subscript 1 refers to the patch state). Thus, assuming Wl ~ VI
p~/R as in OH3 but destinguishing between the swimming speeds of
the predator when searching for patches (Le. W2 ~ VO pred ) and the
swimming speed (VI pred) when foraging inside patches, the mean
count and the IDC will not change as long as Wl » W2. But, of
course, for VOpred > VIpred the average time spend in patches will
exceed the average time spend on interpatch travel.
Mechanisms by which a predator is able to prolong its stay in
patches will decrease WI and hence cause an increase in both the
prey encounter rate and in IDC (as long as Wl > W2). These
considerations are but one example of incorporating behaviour
into the IPP model. Other examples include the effects of mobile
prey organisms and of turbulence on the contact rates. Some of
the principal effects here can be examined by changing the speed
of the predator. The effects of patchiness on encounters with
predators operating with different feeding behaviours such as
ambush feeders and pause-travel predators can be examined as
weIl using the IPP model unit.
In this way I believe the IPP model unit may serve as a welldefined starting point for studying the variability in predatorprey encounter rates in ecologYi a variability which is
important for ensuring sUfficient growth of the survivors and
which at the other end may be responsible directly or indirectly
for a major part of mortality during e.g. early life stages." In
the marine environment the average survivor has most likely not
survived based on the mean encounter rate with prey particles
during early life. In the mortality context the IPP model unit
can be used, for example, to derive analytical expressions for
the probabilities of not encountering food during a critical
time-frame for first feeding fish larvae depending on where the
larvae are located when they are ready to commence feeding.
Addressing many of these questions theoretically will lead to
strong arguments for obtaining the empirical data, which are
needed for further developing and testing the theory. However,
it does required measurements on a scale similar to the one
perceived by the predator in question, which is why further
development of the concept of func~ional heterogeneity seems to
be of importance.
8
•
tt
•
ICES... C.M~".1995."
".
•
.
.,"
I
.'". CM 1995/Q":.10
>
FUNCTIONAL HETEROGENEITY:
•
USING THE INTERRUPTED POISSON"
PROCESS (IPP) MODEL UNIT
IN ADDRESSING HOWFOOD
AGGREGATION MAY AFFECT FISH
RATION
... ... .......
•
• Need for simple but mechanistic models because
predator-prey encounters usually cannot be'
observed and therefore cannot be addressed
without a unifying theory
• Functional heterogeneity = how a predator
perceives and responds to patchy prey
• The IPP building stone provides a stochastic
(individually-based), simple and well-defined
starting point.
q
OH1
•
- INTRODUCTION
.. .... .. ....
Counting process
Interencounter time
pp
EXP
•
POISSON
4r-------~
COMPOUND
POISSON
COMPOUND
EXP
IPP
---t>
•
????????
.. . . . ...
INDEX OF DISPERSION FOR COUNTS
= IDC = VARIANCE TO MEAN RATIO
10
BASIC EXAMPLE
swimming speed
v
visual range
d
ßprey
search volume rate
(encounter rate kerne1)
.. .
~
'.
'r
. .
.
,
.
•
..
•
,,
.
'
7fd2v
average prey density Po
I
..
- ,..,. .
~
.
~
•
=
,
.
An =
encounter rate
-...., -.......
'\
...... ..
...
'"
..
.. ~.
- .
patch volume fraction 'Yl = PolP
intrapatch prey density p = Pol'Yl
. '- ......
prey encounter rate
PATCH ENCOUNTER RATE =
PATCR LEAVING RATE
ßprey' Po
=
W2
W1
1\
=
A=
Anl'Yl
3/4· v . 1]IR
= W211]
= 1 / 4/3·R/V
OH3
MODEL
. ...... .....
IPP transition diagram
A
patch
=
state 1
IPP is stochastically equivalent to a H 2 renewal
process
•
H 2 phase diagram
H 2 process interpretation:
p
1'1
...
-
-+-
different patch concept !
I-p
~
OH4
RESULTS
EXAMPLE:
Mean interencounter time = 1
IDC = variance to mean ratio = 4
•
INDEX OF DISPERSION FOR COUNTS
PREY ENCOUNTER RATE-)
"A./
IDC
•
1 "A
-e-
2 Wz
1
-
-
-
--"-
-
-
-
-
-
W1 )
~
Wz
PATeR LEAVING RATE
PATCR ENCOUNTER RATE
I<'
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